Zeros of lacunary type polynomials

S. Das

Eurasian mathematical journal, Tome 13 (2022) no. 1, pp. 32-43

Résumé

Using Schwarz's lemma, Mohammad (1965) proved that all zeros of the polynomial $$ f(z)=a_0+a_1z+\dots+a_{n-1}z^{n-1}+a_nz^n $$ with real or complex coefficients lie in the closed disc $$ |z|\leqslant\frac{M'}{|a_n|}\text{ if } |a_n|\leqslant M', $$ where $$ M'=\max_{|z|=1}|a_0+a_1z+\dots+a_{n-1}z^{n-1}|. $$ In this paper, we present new results on the location of zeros of the lacunary type polynomial $$ p(z)=a_0+a_1z+\dots+a_pz^p+a_nz^n,\quad p$$ In particular, for $p = n -1$, our first result implies an important corollary which sharpens the above result. Also, we described some regions in which all zeros of $p(z)$ are simple. In many cases, our results give better bounds for the location of polynomial zeros than the known ones.